--- title: "Superdense coding" sort_title: "Superdense coding" date: 2021-03-07 categories: - Quantum information layout: "concept" --- In quantum information, **(super)dense coding** is a protocol to enhance classical communication. It uses a quantum communication channel and [entanglement](/know/concept/quantum-entanglement/) to send two bits of classical data with just one qubit. It is conceptually similar to [quantum teleportation](/know/concept/quantum-teleportation/). Suppose that Alice wants to send two bits of classical data to Bob, but she can only communicate with him over a quantum channel. She could send a qubit, which has a larger state space than a classical bit, but it can only be measured once, thereby yielding only one bit of data. However, they are already sharing an entangled pair of qubits in the [Bell state](/know/concept/bell-state/) $$\ket{\Phi^{+}}_{AB}$$, where $$A$$ and $$B$$ are qubits belonging to Alice and Bob, respectively. Based on the values of the two classical bits $$(a_1, a_2)$$, Alice performs the following operations on her side $$A$$ of the Bell state:
$$(a_1, a_2)$$ | Operator | Result |
---|---|---|
$$00$$ | $$\hat{I}$$ | $$\displaystyle \ket{\Phi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B + \Ket{1}_A \Ket{1}_B \Big)$$ |
$$01$$ | $$\hat{\sigma}_z$$ | $$\displaystyle \ket{\Phi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B - \Ket{1}_A \Ket{1}_B \Big)$$ |
$$10$$ | $$\hat{\sigma}_x$$ | $$\displaystyle \ket{\Psi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B + \Ket{1}_A \Ket{0}_B \Big)$$ |
$$11$$ | $$\hat{\sigma}_x \hat{\sigma}_z$$ | $$\displaystyle \ket{\Psi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B - \Ket{1}_A \Ket{0}_B \Big)$$ |